If one read's Gallian's Abstract Algebra book then they would find that the chapter for Sylow Theorem's is quite hyped up. However, I am unable to understand the big picture of why Sylow subgroups and sylow theorems are important in group theory as a whole. Could some explain the big picture in simple terms?
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7A lot of ink has been spilled on this topic. See here, here, and here to start. The tl;dr, though, is that sylow theorems place serious constraints over what kinds of finite groups can exist. This is interesting, at least to me, because it means there are serious restrictions on what kinds of symmetry can exist! – Chris Grossack Mar 30 '22 at 02:47
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3I don't believe I repeated a link. Though the titles are similar. Also, you asked why "sylow theorems are important in group theory as a whole", and these links give plenty of reasons the sylow theorems are important in group theory. See, for instance, Greg Kupergerg's (excellent) answer at the first link in my earlier comment, but here is a direct link to the answer for convenience – Chris Grossack Mar 30 '22 at 02:57
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You are right. I was mistaken. – Clemens Bartholdy Mar 30 '22 at 02:58
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They are fundamental tools if you want to classify finite groups. – J. De Ro Mar 30 '22 at 14:28
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For a tourist to a new country it is important to know what are the scenic places are. When studying groups one wants to know what its subgroups are.
Lagrange's theorem is the first theorem in this regard: for a subset to be a subgroup its cardinality should divide the cardinality(order) of the group. This is applicable to ALL groups without exception.
On the other hand when taking a divisor of the order of the group, like 6 for the alternating group $A_4$ there is no subgroup of that order.
So it was considered important to know which divisors of the order of the group, guarantee existence of subgroups of that order. Sylow theorem is one such theorem in that direction, and is applicable to ALL groups.
For special kind of groups such as abelian groups it is always possible to find subgroups of any desired order dividing the order of the full group.
The beauty of Sylow theorem is that it also says if we find more than one subgroup of that order they will be conjugate subgroups.
This means, in case in a group of order $810$ we find two subgroups of order $81$, with one of them cyclic, the other can't be direct product of two cyclic groups of order $9$.
P Vanchinathan
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A finite group is both an algebraic object (it has a certain algebraic multiplicative structure) as well as a combinatorial object (one can actually count its elements as a finite set and apply combinatorics and number theory to it). The world-famous Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Mejdell Sylow (1832–1918) who published them in 1872, leveraging both aforementioned properties. They provide detailed information about the number of subgroups of fixed prime power order a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in for example the classification of the finite simple groups, or, for that matter, to prove that particular classes of groups are ${not}$ simple. It is good to realize that Sylow theory not only applies to a finite group $G$ itself and all primes dividing its order, but also holds true for all of its subgroups and quotients. This makes it a very powerful tool indeed.
user3840170
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Nicky Hekster
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Thank you for the answer prof. Both answers were equally great, and in the end, I choose the answer based on the user who had lower reputation points. – Clemens Bartholdy Mar 30 '22 at 20:17
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1No problem. Thanks. I am not aiming at the points, but I like to learn you something and apparently I did. Keep asking more great questions! – Nicky Hekster Mar 30 '22 at 20:24